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Journal of Applied Mathematics
Volume 2011, Article ID 951960, 15 pages
http://dx.doi.org/10.1155/2011/951960
Research Article

On the Diamond Bessel Heat Kernel

Department of Mathematics, Faculty of Science, Maejo University, Chiangmai 50290, Thailand

Received 19 July 2011; Accepted 2 November 2011

Academic Editor: Tak-Wah Lam

Copyright © 2011 Wanchak Satsanit. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We study the heat equation in n dimensional by Diamond Bessel operator. We find the solution by method of convolution and Fourier transform in distribution theory and also obtain an interesting kernel related to the spectrum and the kernel which is called Bessel heat kernel.

1. Introduction

The operator has been first introduced by Kananthai [1], is named as the diamond operator iterated times, and is defined by , is the dimension of the space for , and is a nonnegative integer. The operator can be expressed in the following form: where is the Laplacian operator iterated -times defined by and is the ultrahyperbolic operator iterated -times defined by Kananthai [1, Theorem  1.3] has shown that the convolution is an elementary of the operator . That is where is defined by is a complex parameter, is the dimension of , and the generalized function is defined by and the constant is given by the formula The function is called the ultrahyperbolic kernel of Marcel Riesz and was introduced by Nozaki (see [2, page  72]).

Next, Yildirim et al. (see  [3]) first introduced the Bessel diamond operator iterated -times, defined by where , . The operator can be expressed by , where

And Yildirim (see  [4]) have shown that the solution of the convolution form is a unique elementary solution of , that is, where is defined by (2.8) with and is defined by (2.9) with .

It is well known that for the heat equation with the initial condition where is the Laplace operator and is defined by (1.3) and , we obtain as the solution of (1.13). Now, (1.15) can be written in the classical form where is called the heat kernel, where and , see [5, pages 208-209]. Moreover, we obtain as , where is the Dirac delta distribution.

Next, Saglam et al. (see  [6]) have study the following equation, with the initial condition where the operator is named the Bessel ultrahyperbolic operator iterated -times and is defined by (1.4), is a positive integer, is an unknown function, is the given generalized function, and is a constant, and is the dimension of the .

They obtain the solution in the classical convolution form where the symbol is the -convolution in (2.3), as a solution of (1.18), which satisfies (1.19), where and is the spectrum of for any fixed , and is the normalized Bessel function.

Now, the purpose of this work is to study the equation with the initial condition where is a time, is a positive constant, is an unknown function, and is a given generalized function for . We obtain as a solution of (1.7), where and is the spectrum of for any fixed , and is defined by (2.6) with . The convolution is called the Diamond Bessel Heat Kernel, and all properties will be studied in details. Before proceeding, the following definitions and concepts are needed.

2. Preliminaries

The shift operator according to the law remarks that this shift operator connected to the Bessel differential operator (see [3, 5, 7, 8]): where . We remark that this shift operator is closely connected to the Bessel differential operator (see [3, 5, 7, 8]), The convolution operator determined by the is as follows: Convolution (2.3) is known as a -convolution. We note the following properties of the -convolution and the generalized shift operator.(1),(2),(3) If , is a bounded function for all , and , then .(4) From (3), we have the following equality for : .(5).

The Fourier-Bessel transformation and its inverse transformation are defined as follows: where is the normalized Bessel function which is the eigenfunction of the Bessel differential operator. The following equalities for Fourier-Bessel transformation are true (see [3, 5, 7, 8]):

Definition 2.1. Let , . For any complex number , we define the function by

Definition 2.2. Let , and denote by the nondegenerated quadratic form. Denote the interior of the forward cone by . The function is defined by where and is a complex number.
By putting in and taking into account Legendre’s duplication formula for : we obtain and , where

Definition 2.3. The spectrum of the kernel of (1.21) is the bounded support of the Fourier Bessel transform for any fixed .

Definition 2.4. Let be a point in , and denote by the set of an interior of the forward cone, and denotes the closure of .
Let be spectrum of defined by (1.21) for any fixed and . Let be the Fourier Bessel transform of , which is defined by

Lemma 2.5. Given the equation for , where is defined by (1.10). Then, where is defined by (2.3), with . We obtain that is an elementary solution of the operator . That is

Proof. (See [3, page 379]).

Lemma 2.6. Given the equation for , where is defined by (1.11). Then, where is defined by (2.4), with . We obtain that is an elementary solution of the operator . That is

Proof (see [3, Page  379]). From (2.8) with , we obtain as an elementary solution of the equation Now, from (2.17), We can compute from (2.7) as By using the formula we obtain , and Then, we obtain Thus, where .

Lemma 2.7. Let and be the functions defined by (2.8) and (2.9), respectively. Then where and are a positive even number.

Proof . (See [3, pages 171–190]).

Lemma 2.8 (Fourier Bessel transform of operator). Consider where

Proof. (See [4]).

Lemma 2.9 (Fourier Bessel transform of operator). Consider where

Proof. (see [4, 9]).

Lemma 2.10. For , and , we have where is a positive constant.

Proof. (See [6, Lemma  3.1.1] and [10]).

Lemma 2.11. Let the operator be defined by where is the Laplace Bessel operator defined by is the dimension , is a positive integer, , and is a positive constant. Then, is the elementary solution of (2.15) in the spectrum for .

Proof. Let , where is the elementary solution of and is the Dirac-delta distribution. Thus, Applying the Fourier Bessel transform, which is defined by (2.4) to the both sides of the above equation and using Lemma 2.7 by considering , we obtain Thus, we get where is the Heaviside function, because holds for .
Therefore, which has been already given by (2.7). Thus, from (2.5), we have where is the spectrum of . Thus, we obtain as an elementary solution of (2.15) in the spectrum for .

Definition 2.12. We can extend to by setting

Lemma 2.13. Let us consider the equation with the initial condition where is the operator iterated -times and is defined by is the dimension , is a positive integer, is an unknown function for , is the given generalized function, and is a positive constant. Then, is a solution of (2.43), which satisfies (2.44), where is given by (2.35).

Proof. Taking the Fourier Bessel transform, the both sides of (2.43), for and using Lemma 2.9, we obtain Thus, we consider the initial condition (2.44), then we have the following equality for (2.47): Here, if we use (2.4) and (2.5), then we have where . Set We choose , to be the spectrum of , and, by (2.35), we have Thus, (2.49) can be written in the convolution form Moreover, since exists, we can see that Thus, for the solution of (2.43), we have which satisfies (2.44). This completes the proof.

3. Main Results

Theorem 3.1. Given the equation with the initial condition where is a given generalized function for , is an ultrahyperbolic Bessel operator, and is the Diamond Bessel operator defined by (1.11) and (1.9), respectively. Then, we obtain as a solution of (3.1), where is given by (2.9) with and is given by (2.35).

Proof. Equation (3.1) can be written in the form Let . Thus, the above equation can be written as
We can solve the above equation by applying the -dimensional Fourier Bessel transform to both sides of (3.5). By Lemma 2.7, we obtain By convolving both sides of (3.6) by function , we obtain By properties of convolution, By Lemma 2.5, we obtain or is a solution of (3.1). As shown in (3.5) and by the continuity of convolution,

Theorem 3.2 (The properties of the Diamond Bessel Heat Kernel ). (1) exists and is tempered distribution.
(2) the space of continuous function and infinitely differentiable.
(3) .
(4) .

Proof. (1) Since and are tempered distribution with compact support. Thus, exists and is a tempered distribution.
(2) We have since is infinitely differentiable and .
(3) By the continuity of the convolution, Thus,
(4) Since thus where is defined by (2.35).

Acknowledgments

The authors would like to thank The Thailand Research Fund and Office of the Higher Education Commission and Maejo University, Chiang Mai, Thailand, for financial support and also Professor Amnuay Kananthai, Department of Mathematics, Chiang Mai University, for the helpful of discussion.

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